Problem 20 - Algebra A Combined Function

Question

Use a calculator to approximate each square root to three decimal places. Check to see that each approximation is reasonable. $$ \sqrt{11} $$

Step-by-Step Solution

Answer

The square root of 11 is approximately 3.317.

Step 1: Use a Calculator

Enter 11 into your calculator and press the square root function key to approximate the square root of 11. You should get a decimal result. Round this result to three decimal places.

Step 2: Round to Three Decimal Places

The calculator should display something close to 3.31662479036. However, we only need the value up to three decimal places. Therefore, the proper rounding of this number is 3.317.

Step 3: Verify the Reasonableness

Compare the approximate square root, 3.317, with values you know. Since 3 squared is 9 and 4 squared is 16, it's reasonable that the square root of 11 would be between 3 and 4, supporting the reasonableness of our approximation.

Key Concepts

Calculators in Mathematics

Calculators are indispensable tools in mathematics that simplify complex calculations. When it comes to approximating square roots, calculators can quickly provide you with a decimal approximation of a number that would otherwise be tedious to compute manually. Enter the number whose square root you want to find into the calculator and press the square root button, often denoted as \( \sqrt{} \). This will give you a decimal value that approximates the root. Once you have the decimal, it’s easier to verify its accuracy and reasonableness by comparing it to known square roots.
Calculators not only save time but also improve precision, ensuring that calculations are accurate to several decimal places. This precision is particularly important in scientific fields where minor errors can have significant consequences. Even though calculators are exceptionally useful, it's essential for students to understand the underlying concepts to better interpret and use these results effectively.

Rounding Decimals

Rounding decimals is a mathematical process where you reduce the number of digits in a decimal to make the number easier to read and work with. This is crucial when working with calculator results, as display screens usually show more digits than necessary. To round a number to three decimal places, find the third decimal place and look at the digit immediately following it.
If this digit is 5 or greater, round up the third decimal place. If it is less than 5, keep the third decimal place the same. This systematic approach to rounding helps maintain accuracy and makes it easier to communicate results in simpler, more understandable terms. For example, when approximating \( \sqrt{11} \), we round 3.31662479036 to 3.317.

Comparing Square Roots

Comparing square roots involves understanding number properties and using estimations to check if your solutions are reasonable. One way to do this is by identifying perfect squares around the number you’re evaluating. For instance, if you know \(3^2 = 9\) and \(4^2 = 16\), then \( \sqrt{11} \) must be between 3 and 4.
Establishing these boundary points allows you to see that your approximation is logical. In the case of approximating \( \sqrt{11} \) to 3.317, knowing it falls between the square roots of 9 and 16 reaffirms your estimation is reasonable. This method of comparing helps in assessing the validity of your results without needing precise calculations and reinforces your understanding of how square roots of non-perfect squares behave.